The impact of stochastic microstructures on the macroscopic fracture properties of brick and mortar composites
Introduction
The ability to connect local microstructural features with macroscopic mechanical properties is a central challenge in materials development, as such connections effectively define ‘processing targets’ associated with acceptable levels of heterogeneity. This challenge is particularly acute for quasi-brittle composites that exploit ordering of microstructural features, as local disruptions to ordering can serve as defects that limit global performance [1], [2], [3], [4]. Bio-inspired ‘brick and mortar’ composites are an excellent example of this [5], [6], [7], [4], [8], [9]; the ideal microstructure will have perfectly aligned stiff features and achieve toughness through perfectly overlapping features that spread damage (or equivalently, provide bridging over large length scales).
However, the physical reality is that most high through-put processing routes invariably introduce heterogeneous distributions of brick size and therefore overlap distances, which limit the effectiveness of the microstructure. As such, the macroscopic performance of these composites is less than optimal, as real materials (both natural and synthetic) exhibit imperfections at different length scales, including large defects/pores [10], [11], [12], [13], [14], [15], [16], [17], distributions in brick sizes and shape [18], [19], [20], and distributions in mortar properties (e.g. strength) from interface to interface. These features not only reduce the effective macroscopic mechanical properties of the material (broadly defined), but they can also lead to substantial variability in properties from one test specimen to the next [21], [22], [23], [24], [5], [6].
The stress–strain response of brick and mortar composites has been modeled extensively in a deterministic sense via analytical models as well as conventional finite element simulations [25], [26], [27], [28], [29], [22], [30], [31], [32]. However, even for uniform brick arrangements, direct links between microstructure and the resulting fracture toughness (arguably their most desirable property) are limited [6], [33], [34], [29]. It should be noted that while the work-to-failure during uniform deformation is related to toughness, the presence of non-uniform deformation at the tip of a crack leads to important differences [33]. To our knowledge, the impact of non-uniform and statistically-defined brick arrangements on macroscopic fracture toughness has not been analyzed. As a result, there is a significant need for a quantitative understanding of the link between statistical distributions in microstructure (e.g., brick sizes, shape, mortar properties) and the resulting statistics of fracture properties.
Previous analyses of the impact of heterogeneous brick arrangements on uni-axial behavior have provided insights that are relevant to the present explicit study of fracture toughness. These works defined representative volume elements (RVEs) of brick arrangements within finite element models with embedded cohesive zones [18], [35]. For example, Barthelat et al. studied the effect of statistical size and shape distributions in the microstructure using a large RVE containing several hundreds of tablets generated via Voronoi tiling. Their results indicated important differences in composite stress–strain response of a microstructure with statistical variations when compared to that of a ‘perfect’ microstructure, and they are in excellent agreement with experiment [18]. In the work of Bekah et al., the authors also implemented statistics within the microstructure by introducing a random variation in tablet length and offset [35]; the results indicated a reduction in both strength and work to failure as a result of the variation. Moreover recently, statistical strength predictions in scaffolded ceramic structures (analogous to natural nacre) have begun to emerge [30]; these calculations have enabled predictions of strength probability distributions in periodic materials, and can be generalized for arbitrary geometry and loading.
The objective of this work is to elucidate the statistical characteristics of macroscopic fracture properties that result from statistical variations in local microstructural features, using as a reference the ‘perfect’ material (e.g., one with no variation in microstructure). The current models exploit the highly efficient GPU-based computational approaches developed by Pro et al. and Lim et al. [33], [34] to conduct a broad number of virtual fracture tests. The results confirm some of the trends suggested by previous works, yet provide more complete statistical links between parameters describing microstructural heterogeneity and macroscopic fracture properties. The simulations utilize virtual specimens comprising several hundreds of thousands of bricks, with sizes determined via sampling from a statistical distribution. The approach explicitly tracks individual motions of each brick (all of different sizes) and predicts crack evolution through the rupture of cohesive zones bonding the bricks. The efficiency of the numerical method allows a large number of virtual tests, such that the statistics of the response can be well quantified.
The results offer new insight into the design of synthetics, illustrating the role of statistics in the microstructure on both the average and standard deviation in initiation fracture toughness, as well as the resulting damage and stress distributions ahead of a dominant pre-crack. The simulation results give key insight into the sensitivity of the composite to local variations in brick size, in terms of the computed average and variance from a statistically significant subset of virtual fracture tests.
Section snippets
Microstructural idealizations
As illustrated in Fig. 1, the material is modeled using an idealized microstructure consisting of an overlapping grid of comparatively stiff bricks bonded together by thin, compliant mortar sections. For many synthetic composites the bricks can be modeled as rigid bodies with only rotational and translational degrees of freedom, as described in detail in Pro et al. [33]. The mortar, presumed to be present in low volume fraction, is idealized with a piece-wise cohesive zone law (shown
Energy release rates and fracture toughness
Control simulations were first performed with a set of perfectly stacked bricks with 50% overlap, all of equal size, for several different brick aspect ratios (, and 5.25 for all brick indices). The effect on the overall crack driving force (normalized appropriately by the work-to-failure of a given interface, ) and inferred toughness was studied for different brick aspect ratios as a function of the Monte Carlo seed. The results confirmed that the influence of the Monte
Concluding remarks
The results presented in this paper show a powerful application of the method presented by Pro et al. [33] and Lim et al. [34], illustrating general capabilities to simulate large specimens that contain distributions in brick sizes; further, the speed enables one to simulate enough fracture tests to obtain meaningful statistics on the macroscopic fracture properties. The key results and their implications are outlined below:
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Variability in brick sizes decreases the effective elastic
Acknowledgments
This research was financially supported through grant DMR-1105672 from the National Science foundation, grant W911NF-09-0001 through the Institute for Collaborative Biotechnologies from the US Army Research office, and grant DOE/DE-SC0008975 from the Department of Energy. Additional computational resources were provided by the Center for Scientific Computing at the CNSI and MRL: an NSF MRSEC (DMR-1121053) and NSF CNS-0960316. The content and information presented do not reflect the position or
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2020, Journal of the Mechanical Behavior of Biomedical MaterialsCitation Excerpt :Nacre, the inner layer of mollusk shells, has received considerable research attention in the past few decades due to its excellent combination of high strength and superior toughness (Barthelat et al., 2007, 2011; Kuhn-Spearing et al., 1996; Menig et al., 2001; Smith et al., 1999). These remarkable mechanical performances of nacre are achieved through its complex brick-and-mortar-type hierarchical microstructure organized over the nanoscale and above (Abid et al., 2018; Al-Maskari et al., 2017; Askarinejad et al., 2018; Dutta et al., 2013; Gao et al., 2019; Henry et al., 2018; Pro et al., 2015; Ni et al., 2015). The brick constituent, comprising a layer of microscopic aragonite polygonal tablets arranged in a highly regular staggered form, is bonded together by 20–30 nm thick layers of organic material, serving as “adhesive interfaces” (Rim et al., 2011).